Advances in Energy and Power 4(3): 23-34, 2016 DOI: 10.13189/aep.2016.040301 http://www.hrpub.org

Design of Modified Maiden Power System StabilizerUsing Cuckoo Search Algorithm

D. K. Sambariya

Department of Electrical Engineering, Rajasthan Technical University, Kota, 324010, India

Copyright c©2016 by authors, all rights reserved. Authors agree that this article remains permanently

open access under the terms of the Creative Commons Attribution License 4.0 International License

Abstract This article presents an improved maiden powersystem stabilizer (PSS) for enhancement of small signalstability of a power system. The free coefficients of proposedPSS are determined using optimization technique with thecuckoo search algorithms (CS-PSS). The performance of theCS-PSS is validated on single-machine infinite-bus powerand extended to a multi-machine power system. Theseresults are compared to the newly introduced maiden PSSstructure and found superior in terms of settling time andperformance indices.

Keywords Power System Stabilizer, Single-machineinfinite-bus power system, Two-area Four-machine Ten-busPower System, Cuckoo Search Algorithm, Maiden PSS

1 IntroductionThe energy issue is one of the important challenges in

modern scenario. It consists of the power generation, trans-mission and distribution of the energy to the end users. Theresulting network is a large and complex in sense of analysisand operation. On occurrence of sudden load changes andfaults on the system, results to small signal oscillations in therange of 0.2 Hz to 3.0 Hz. These oscillations tend to die-out automatically, but some of these may persist for a longertime causing power transfer impossible over the weak trans-mission lines [1].

In early phase of 1960s, the fast acting, high-gain auto-matic voltage regulators (AVR) were applied to the generatorexcitation system which in-turn invites the problem of lowfrequency electromechanical oscillations in the power sys-tem. The device connected to generator excitation to controlthe oscillations were termed as power system stabilizer. Itadds a stabilizing signal to AVR for modulating the genera-tor excitation such as to create an electric torque componentin phase with rotor speed deviation, which increases the gen-erator damping [2].

These stabilizers were designed to make system oscilla-tion free with different structural designs and/or control tech-

niques. The early development of PSS were lead-lag andwere called as conventional power system stabilizer. Sim-ilar to CPSS; a Proportional-Integral-Derivative (PID) con-troller may be connected to modulate the signal of the AVR todamp-out the small signal oscillations. The conventional tun-ing method of the PID gains is based on as Zeigler/Nichol’smethod, gain-phase margin method, Cohen/Coon pole place-ment, gain scheduling and minimum variance methods. Re-cently, a new PSS structure is proposed in [3], as similar toCPSS and PID based PSS. However, these methods sufferfrom some limitations as (a) extensive methods to set gains,(b) difficulty to deal with gains for a large, complex and non-linear power system, and (c) poor performance in a closedloop because of changing conditions [4, 5].

The design of power system stabilizer is explored usingfuzzy logic controller [6, 7]. It have been considered formulti-machine models of power system in [8, 9]. The roleof membership function in the design of PSS is examinedin [10] and with different de-fuzzification methods in [11].The robust fuzzy PSS is presented in [12]. The role of mem-bership funtion based on linguistic variables are examined in[13].

To mitigate the shortcomings of these conventional meth-ods much optimization based algorithms have been pro-posed. The methods available in literature are as Tabu search[14], Evolutionary algorithm [15], the Differential Evolu-tion (DE) algorithm [16], Simulated Annealing [17], GeneticAlgorithm [18], particle swarm optimization [19], an iter-ative linear matrix inequalities algorithm [20], Combinato-rial Discrete and Continuous Action Reinforcement Learn-ing Automata (CDCARLA) [21], Bacteria Foraging Opti-mization (BFO) Algorithm [22], Bat Algorith (BA) as in[23, 24, 25, 26, 27], Harmony Search algorithm (HSA) as in[28, 29, 5], Fire fly algorithm (FFA) [30] and other than theoptimization, some artificial intelligence based, techniquessuch as type-1 Fuzzy logic based PSS [31, 29, 28, 32, 33], In-terval type-2 Fuzzy logic based PSS [34, 35, 36], ANN [37]etc. are ready for use in the design of PSS.

The above optimization methods work well but fail withthe objective function as highly epistatic with a large num-ber of parameters. To such objective function, these methods

24 Design of Modified Maiden Power System Stabilizer Using Cuckoo Search Algorithm

may give degraded results with a large computational burden.Meta-heuristic algorithms possess two important charac-

teristics like intensification (or exploitation) and diversifica-tion (or exploration) is considered as upper-level methodsfor the optimization. Genetic algorithm [38, 39] and parti-cle swarm optimization [40, 19, 41] are the typical types ofmeta-heuristic algorithms for global optimization in the de-sign of a power system stabilizer.

Yang and Deb in 2009 [42], have introduced a promisingnature -inspired metaheuristic algorithm called as Cuckoosearch (CS) and extended to engineering optimization in[43] and multi-objective optimization in [44, 45, 46]. Civi-cioglu and Besdok (2013) [47], have introduced a concep-tual comparison of cuckoo search with differential evolu-tion (DE), particle swarm optimization (PSO), artificial beecolony (ABC) and suggested that differential evolution andcuckoo search algorithms provide more improved resultsthan ABC and PSO. Gandomi et al. (2013) [48], provided amore extensive comparison study for solving various sets ofstructural optimization problems and concluded that cuckoosearch obtained improved results than other algorithms suchas PSO and genetic algorithms (GA). Among the diverse ap-plications, an interesting performance enhancement has beenobtained by using cuckoo search in reliability optimizationproblems in [49].

The main concern of this article is to evaluate and modifythe maiden PSS structure proposed in [3]. The maiden PSSstructure is modified with the knowledge of modern controltheory as required for system to be stable resulting additionof non-zero in the numerator part of the compensator. Thefree elements of such modified maiden PSS are optimizedusing CSA (PSS: Proposed) and compared to the maiden PSS(PSS: Falehi) by connecting both controllers to SMIB andmulti-machine power system.

In the organization of paper, the problem is formulated insection 2. The Cuckoo search algorithm which is used to op-timize the PSS controller parameters is introduced in section3. The performance analysis is carried out in section 4, forsingle-machine infinite-bus power system and multi-machinepower system model. Lastly the analysis is concluded in sec-tion 5, followed by appendix and references.

2 Problem Formulation

The general representation of a power system using non-linear differential equations can be given by

X = f(X,U) (1)

Where, X and U represents the vector of state variablesand the vector of input variables. As in [29], the powersystem stabilizers can be designed by use of the linearizedincremental models of power system around an operatingpoint. The system representation based on differential equa-tions and used data is given in [23]. The state equations of apower system can be written as

∆X = A∆X +BU (2)

2.1 SMIB power system

The schematic diagram of the single-machine connected toan infinite-bus (SMIB) through a transmission line is shownin Fig 1. It includes the generator, AVR and excitationsystem, PSS, transmission line and the infinite-bus. Theinfinite-bus system is the representation of a large intercon-nected power system which is generally represented by theThevenins equivalent.

Figure 1. The schematic representation of SMIB system

Figure 2. Representation of Heffron-Phillip model of SMIB power system

The excitation system and the AVR system are connectedto the generator as in Fig. 1. The deviation in the generatorspeed is sensed and applied as input to PSS. The output of thePSS is applied to excitation system to modulate the signal.To operate power system in synchronism an adequate damp-ing torque is required. The excitation with AVR system un-able to meet requirement of an adequate damping, therefore,to provide extra damping using subsidiary excitation controlthe PSS have been developed as in [31, 1]. The linearizedmodel of SMIB was the result of a first serious investigationby DeMello and Concordia in 1969 [50]. In system repre-sentation by Eqn. 2, A is the system matrix with order as4×4 and is given by δf/δX , while B is the input matrixwith order 4×1 and is given by δf/δU . The order of statevector is 4×1, the order of is 1×1. Here, the well known

Advances in Energy and Power 4(3): 23-34, 2016 25

Heffron-Phillip linearized model and the connection to FPSSwith scaling factors is shown in Fig. 2 [16].

2.2 Two-area four-machine power system

The schematic diagram of the four-machine ten-bus powersystem is shown as in Fig. 3. The analysis of the sys-tem can be carried out by simultaneous solution of equationsconsisting of synchronous machines with excitation systems,prime movers, dynamic and static loads, transmission linenetwork, and other devices like static VAR and HVDC con-verters based compensators. The dynamics of generator ro-tors, prime movers, excitation, and other related devices arebeing represented by differential equations. Thus, the com-plete multi-machine model consists of large numbers of or-dinary differential equations (ODE) and algebraic equations[28, 29]. These are linearized about an operating point (nom-inal) to derive a linear model for the small signal oscillatorybehaviour of power systems. The range of variation in oper-ating point can generate a set of linear models correspondingto each operating point/condition.

Figure 3. Representation of line diagram for fou-machine ten-bus powersystem

Figure 4. Representation of Heffron-Phillip model for multi-machine con-figuration of power system

The state equations of a power system, consisting N num-ber of generators and Npss number of power system stabiliz-ers can be written as in Eqn. 2. Where, A is the system ma-trix with order as 4N × 4N (16× 16) & is given by δf/δX ,

while B is the input matrix with order 4N × Npss (16 × 4)and is given by δf/δU . The order of state vector is 4N × 1(16 × 1), the order of is Npss × 1 (4 × 1). Here, the wellknown Heffron-Phillip linearized model is used to representthe large multimachine power system as in Fig. 4 [29].

2.3 PSS proposed in Falehi [3] and Proposed

The general requirement of a power system stabilizer tocompensate the developed phase lag in between excitation in-put and air-gap torque, therefore, a phase compensator blockis needed. In 2013 [3], Falehi have proposed a new struc-ture of PSS as in Fig. 5, but it lakes with the provision ofproper phase compensation in compensation block. There-fore, in Fig. 6, proper phase compensation is introduced bynon-zero in the phase compensation block. Derivative andintegral blocks are kept same as in Falehi PSS [3]. In case ofFalehi PSS, there are four parameters (Tc, Ac, Ki, Kd) to beoptimized by cuckoo search algorithm, while these are five(Tp, Tc, Ac, Ki, Kd)in the proposed new PSS structure as inFig. 6.

Figure 5. PSS structure as in [3]

Figure 6. Proposed PSS structure

2.4 Objective function

To increase the system damping to electromechanicalmodes, of the power system model five different objectivefunctions are considered. The problem constraints are as theparameters of the controllers connected to the power system.The unknown parameter bounds are considered as in Eqn. 3- 7.

Tminp ≤ Tp ≤ Tmax

p (3)

Tminc ≤ Tc ≤ Tmax

c (4)

Aminc ≤ Ac ≤ Amax

c (5)

Kmini ≤ Ki ≤ Kmax

i (6)

Kmind ≤ Kd ≤ Kmax

d (7)

Typical ranges of the optimized parameters are 0.1 ≤Tp ≤ 1.5, 10 ≤ Tc ≤ 30, 10 ≤ Ac ≤ 20, 0.01 ≤ Ki ≤ 0.5,200 ≤ Kd ≤ 300, respectively. The above parameters of

26 Design of Modified Maiden Power System Stabilizer Using Cuckoo Search Algorithm

the controller are determined by HS algorithm under the oneobjective function as describe following.

J |SMIB =

t=Tsim∫t=0

|∆ω(t)|2dt (8)

J |MM =

t=Tsim∫t=0

4∑i=1

|∆ωi(t)|2dt (9)

3 Cuckoo Search AlgorithmThe application of CS algorithm in the field of optimiza-

tion has received appreciable attention. It has been modifiedtime to time according to problem requirements. It have beenmodified to deal with mult-objective problems by Yang andDeb [44] and proposed a modified CS algorithm by Waltonet al. in [51].

As the Cuckoos lay their eggs in the nest of other birds andrespective host birds take care of the cuckoos chicks [52]. Itis mainly inspired by the obligate brood parasitism of cuck-oos by laying their eggs in the nests of other host birds. Theinfringing cuckoos are in direct contest with the host birds.The host bird discovers the eggs of other birds and may throwthese out of nest or may construct another nest elsewhere.The Parasitic cuckoos generally selects a nest in which thehost bird just laid its own eggs [52]. The Cuckoo eggs gen-erally hatch somewhat earlier than their host eggs [53]. Assoon as, cuckoo chick is hatched starts to evict y blindly pro-pelling the eggs out of the nest to reduce the share of food.Cuckoo chick starts to mimic the voice call of host chicks togain more opportunity of feeding [52, 54].

An algorithm provides a set of output variables on appli-cation of input variables. An optimization algorithm gener-ates/produces a new set of solution xt+1 to a given problemfrom a given solution xt at time t or iteration.

xt+1 = A{xt, p(t)} (10)

Where, the new solution vector xt+1 is nonlinearlymapped through A to given d-dimensional vector xt. Letthe variables of the problem are k and are represented asp(t) = p1, p2, ..., pk which may be time dependent and canbe tuned by A. Let an optimization problem is S with statesas ψ then according to pre-define criterion D, the optimalsolution xos selects the desired states as φ as in Eqn. 11.

S(ψ)A(t)−→ S{φ(xos)} (11)

Thus, the final found/converged state φ represents to an op-timal solution of the problem of interest. Here, the systemstates are selected in the design space by running the opti-mization algorithm A. Thus, the performance of the algo-rithm is depended /controlled by the initial solution xt=0, theparameters p, and stopping criterion.

3.1 Procedural steps

The Cuckoo search algorithm is based on the brood par-asitism of some cuckoos such as the ani and Guira and is

enhanced by use of Levy flights [55], not just by simpleisotropic random walks. The Cuckoos are special birds notonly because of the beautiful sounds but also because of theiraggressive reproduction strategy. Cuckoos engage the obli-gate brood parasitism by laying their eggs in the nests of otherhost birds. The ani and Guira as the species of cuckoos usedto lay their eggs in other birds nests and they may removeothers eggs to increase the hatching probability of their owneggs. It is necessary to make assumptions as followings:

Assumptions

• At a time each cuckoo lays one egg and dumps it in arandomly selected nest

• The nests with high-quality eggs are selected and beingcarried over to the next generations

• The available number of nests (of hosts) is kept fixed(as n), and the probability of cuckoo egg detection bythe host bird is fixed as Pa ∈ [1, 0]. As above, the hostbird may get rid of the egg or may even abandon the nestto build a new nest i.e a fraction Pa of the n host neststhat are replaced by new nests [52].

Further, as an implementation, it should be assumed thatthe solution refers to an egg in a nest, and each cuckoo can layonly one egg. Thus, there is no distinction between cuckoo,egg or nest because as each nest consists one egg which cor-responds to one cuckoo. CS algorithm uses a combinationof a local random walk (for local search) and the global ran-dom walk (for global search) and is controlled by a switchingparameter Pa.

Local random walk: Let two different solutions selectedby random permutation are as xtj and xtk, Heaviside functionas H(Pa− ∈) , random number drawn from a uniform distri-bution as ∈, and with step size as s. Then, the local randomwalk can be represented as.

xt+1i = xti + αs⊗H(Pa− ∈)⊗ (xtj − xtk) (12)

Here, α > 0 is the step size related to the scales of theproblem of interests. It is generally selected as α = 0.Theproduct ⊗ means entry-wise walk during multiplications.

Global random walk: The global random walk is carriedout by using Levy flights in which the step-lengths are dis-tributed according to a heavy-tailed probability distribution[52]. On completion of large number of steps the randomwalk tends to a stable distribution as compared to its origin.The final solution can be represented by Eqn. 13 as follow-ing.

xt+1i = xti + αL(s, λ) (13)

L(s, λ) =λΓ(λ) sin(πλ/2)

π

1

s1+λ(14)

The Eqn. 13 is the stochastic representation for a randomwalk. The random walk is a Markov chain; whose next loca-tion directly depends on the current location and the tran-sition probability. An appropriate value of new solutionsgenerated by randomization and their locations should be farenough from the best solution (current) to make sure not betrapped in a local optimum [53, 42]. The local search existsabout to 1/4 of the search time (with Pa = 0.25), while globalsearch exists for 3/4 of the total search time.

Advances in Energy and Power 4(3): 23-34, 2016 27

Levy distribution: Levy flights are characterized by infinitemean and variance therefore, CS can explore the search spacemore efficiently as compared to standard Gaussian process.Thus, CS guaranteed global convergence and highly efficient[53, 56, 57].

In Levy flight the step-lengths are distributed according tothe probability distribution as in Eqn. 15, which provides arandom walk while the random step length is drawn from aLevy distribution for 1 ≤ λ ≤ 3 [53].

Levy(u) = t−λ (15)

Improved cuckoo search: As above the α introduced in theCS is to find locally improved solutions, while Pa and λ tofind global solution. In tuning of solution vectors; the param-eters Pa and α plays a vital role. In original CS, Pa and αare kept fixed and cannot be altered during new generations,therefore, the number of iterations kept large to get optimalsolutions. With large value of Pa and small value of α, theconvergence speed is high but unable to find required solu-tions. To mitigate the problem of adjusting the value of Paand α, these are considered as variables in improved CS. Thevalues of Pa and α must be large enough to make capable thealgorithm to increase the diversity of solution vectors duringearly generations and decreased in final generations to resultin a better fine-tuning of solution vectors. Thus, Pa and αare dynamically changed with the number of generation andexpressed in Eqn. 16 - Eqn. 18, where NI and gn are thenumber of total iterations and the current iteration, respec-tively [52].

P gna = Pa,max −Pa,max − Pa,min

NIgn (16)

α(gn) = αmax × e(c.gn) (17)

c =Ln(αmin/αmax)

NI(18)

The performance of the algorithm may deteriorate by anincrease in the maximum value of α as in [52], therefore,the suitable values are 0.005 ≤ Pa ≤ 1.0 and 0.05 ≤ α ≤0.5. The considered values of Pa and α are 0.25 and 0.25,respectively. The Cuckoo Search is shown in Algorithm 1.

4 System response and discussion4.1 SMIB power system

4.1.1 Controller parameter optimization

In order to assess effectiveness, the proposed CS-PSS al-gorithm is programmed in MATLAB R2011b environmentand executed on Intel (R) Core (TM) - 2 Duo CPU T64002.00 GHz with 3 GB RAM, 32-bit operating system. The pa-rameters of the algorithm used for simulation are: n = 25,Pa = 0.25 and Iteration as 200 as in Algorithm 1. The plant(SMIB power system) operating at nominal operating con-dition (where in Xe = 0.4p.u. and P = 1.0p.u.) is con-sidered for optimal tuning of PSS parameters as proposed in[28]; subjected to the ISE minimization based objective func-tion with the parametric bounds such as 0.1 ≤ Tp ≤ 1.5,

Algorithm 1 Cuckoo search algorithm for tuning parametersof conventional power system stabilizer

1: procedure OBJECTIVE FUNCTION F (X), X =(X1, X2, ..., Xd)

T (minimization of objective function;where Xd is the number of free Coefficients of CPSS)

2: Initialize a population of a host nest, xi, (i = 1, 2, ..., n);selected as n =25, lower and upper bound are defined invector

3: for i = 1 : n, nest(i, :)=Lb+(Ub − Lb). ∗rand(size(Lb)) do

4: end for5: while iter < Maximumgenerations do6: Get a cuckoo (say i) randomly & generate a new solution

by levy flights as in Eqn. 15. Evaluate its quality / fitnessFi,Choose a nest among n (say j) randomly.

7: if Fi < Fj then8: Replacing j by the new solution i.e. replacing with min-

imum function value.9: end if

10: Abandon a fraction (Pa) of worse nests and generate(Pa ∈ [0, 1], as 0.25 in Eqns. 16 - 18 new solutionsat new location by Levy flights (as in Eqn. 15)

11: keep the best solutions(bestnest) i.e. nests with qual-ity solutions rank the solutions and find the current best(fmin);

12: iter = iter + 1; (update iteration counter)13: Fcs(iter, :) = fmin; save Fcs.mat {to plot fitness func-

tion or value at each iteration[200× 1] as in Fig. 8 - Fig.9}

14: Pcs(iter, :) = bestnest; save Pcs.mat {Parameters orvalue at each iteration}

15: end while16: post process results(fmin, bestnest) and visualization17: end procedure

10 ≤ Tc ≤ 30, 10 ≤ Ac ≤ 20, 0.01 ≤ Ki ≤ 0.5 and200 ≤ Kd ≤ 300. The scheme of optimization is shownin Fig. 7 and the performance of cuckoo search in terms offitness function variation is shown in Figs. 8 - 9. The op-timized parameters for both PSSs (Falehi PSS and ProposedPSS) at nominal operating condition are enlisted in Table 1.The fitness function value at 200th iteration for Falehi PSSand proposed PSS are as 8.352 × 10−4 and 7.053 × 10−4,respectively.

Figure 7. Scheme of parameter optimization using cuckoo search algorithmfor PSS: Falehi [3] and PSS: Proposed

28 Design of Modified Maiden Power System Stabilizer Using Cuckoo Search Algorithm

Table 1. Optimized parameters using cuckoo search algorithm for PSS (Proposed) and PSS (Falehi) [3]

Structure Tp Tc Ac Ki Kd

PSS: Proposed 0.3991 22.9353 16.0501 0.0027 300PSS: Falehi [3] - 14.5637 10 0.0187 200

Figure 8. Performance of cuckoo search algorithm in parameter optimiza-tion for Falehi PSS Structure [3] in SMIB system

Figure 9. Performance of cuckoo search algorithm in parameter optimiza-tion for proposed PSS Structure in SMIB system

4.1.2 Performance analysis

The considered power system is subjected to fault at 5 sec-onds (persists up to 0.1 second i.e. cleared at 5.1 seconds)and the performance of both PSS structures in terms of gen-erator speed, control voltage, voltage behind transient reac-tance, air-gap electric torque, power angle and terminal volt-age is compared fin Fig. 10 - 15. It is clear that the systembehaviour without PSS is unstable, while it is being stabi-lized using either PSS structure. The recorded settling timewith PSS [3] is 15.1 seconds and with PSS (Proposed is 8.2seconds) as shown in Fig. 10, results heavy performance im-provement with proposed PSS. The other signal variationswith proposed PSS structure, such as control voltage, volt-age behind transient reactance, air-gap electric torque, powerangle and terminal voltage shown in Fig. 11 - Fig. 15, re-spectively are also settled to steady state appreciably earlierthan that with PSS structure as in [3].

Figure 10. Plot of SMIB response with PSS structure proposed as in [3] andproposed PSS structure for speed deviation

Figure 11. Plot of SMIB response with PSS structure proposed as in [3] andproposed PSS structure for control signal

Figure 12. Plot of SMIB response with PSS structure proposed as in [3] andproposed PSS structure for internal voltage

Advances in Energy and Power 4(3): 23-34, 2016 29

Figure 13. Plot of SMIB response with PSS structure proposed as in [3] andproposed PSS structure for electric torque

Figure 14. Plot of SMIB response with PSS structure proposed as in [3] andproposed PSS structure for change in angle

Figure 15. Plot of SMIB response with PSS structure proposed as in [3] andproposed PSS structure for terminal voltage

4.2 Two-area four-machine ten-bus power system

4.2.1 Controller parameter optimization

Considering same parameters of CS algorithm as in pre-ceding section and the actuating data for line diagram in Fig.3 as in [29, 28] equipped with four controllers to four gen-erators are optimized. The performance of CSA in terms offitness function (J for multi-machine) variation is recordedas in Fig. 16 and Fig. 17. The fitness function value at the200th iteration with PSS structure as in [3] is 0.2271 and withthe PSS (proposed) is 0.0511.

The higher value of fitness function with PSS [3] rep-resents its premature optimization at 20th iteration and onwards. The optimized parameters with both controllers are

Figure 16. Performance of cuckoo search algorithm in parameter optimiza-tion for Falehi PSS Structure [3] in Two-Area System

Figure 17. Performance of cuckoo search algorithm in parameter optimiza-tion for proposed PSS Structure in Two-Area System

enlisted in Table 2. The speed signal for all four generators(Gen-1 to 4) without PSS, with PSS [3] and with PSS (Pro-posed) is recorded in Fig. 18 - Fig. 21, respectively. It isclear from these figures that all generators without PSS showunstable behaviour and response with both PSSs as stable.As a comparison the settling time with both PSS structure isrecorded in Table 3 and is clear that the performance withproposed PSS is very encouraging because settling to steadystate quite earlier. The 5th column of Table 3 represents thepercentage improvement (about 86 to 88).

Figure 18. Speed response of two-area power system without PSS, with PSSstructure as in [3] and with PSS (Proposed) for Generator-1

It is very clear from above time domain analysis thatthe performance with proposed PSS outperform the PSS byFalehi, but to have more clearer quantitative analysis three

30 Design of Modified Maiden Power System Stabilizer Using Cuckoo Search Algorithm

Table 2. Cuckoo search based optimized parameters for (a) PSS: Falehi [3] and (b) PSS: Proposed

Controller Genrs Controller Parameters

Tp Tc Ac Ki Kd

PSS: Proposed

Gen-1 0.10 10.00 10.71 0.28 295.82Gen-2 0.31 10.01 10.24 0.06 296.44Gen-3 0.11 12.28 19.76 0.50 200.00Gen-4 1.01 10.00 10.25 0.50 289.79

PSS: Falehi

Gen-1 - 30.00 10.00 0.49 200.00Gen-2 - 20.40 10.00 0.50 299.99Gen-3 - 30.00 10.00 0.50 200.01Gen-4 - 30.00 10.00 0.50 200.07

Table 3. Settling time in seconds for speed response of system without PSS, with PSS: Falehi [3] and with PSS: Proposed

Generator Without PSS PSS:Falehi [3] PSS: Proposed Improved (%)

Gen-1 Unstable 84.83 10.92 87.13Gen-2 Unstable 95.32 11.32 88.12Gen-3 Unstable 71.88 8.461 88.23Gen-4 Unstable 71.60 9.407 86.86

Figure 19. Speed response of two-area power system without PSS, with PSSstructure as in [3] and with PSS (Proposed) for Generator-2

Figure 20. Speed response of two-area power system without PSS, with PSSstructure as in [3] and with PSS (Proposed) for Generator-3

types of performance indices (PIs) are introduced as in Eqn.19 - Eqn. 21 and evaluated as in Table 4.

Figure 21. Speed response of two-area power system without PSS, with PSSstructure as in [3] and with PSS (Proposed) for Generator-4

• ITAE: Integral of the Time-Weighted Absolute Error

ITAE =

Tsim∫0

t |∆ω(t)| dt (19)

• ISE: Integral Square Error

ISE =

Tsim∫0

|∆ω(t)|2dt (20)

• IAE: Integral of the Absolute Error

IAE =

Tsim∫0

|∆ω(t)| dt (21)

where, Tsim is the simulation time of the system consid-ered as 100 seconds. It is found that the least value for allPI’s associated to PSS (proposed) resulting to guarantee thebetter performance as against PSS by Falehi.

Advances in Energy and Power 4(3): 23-34, 2016 31

Table 4. Performance indices (ITAE, IAE and ISE) for speed response with (a) PSS: Falehi [3] and (b) PSS: Proposed

Genr.ITAE IAE ISE

Falehi Prop. Falehi Prop. Falehi Prop.

G-1 0.7128 0.0092 0.0326 0.0034 3.0848E-05 4.1870E-06G-2 0.7221 0.0088 0.0326 0.0032 3.0204E-05 3.8272E-06G-3 0.7018 0.0112 0.034 0.0053 4.5987E-05 1.9542E-05G-4 0.5384 0.0120 0.0272 0.0047 3.0717E-05 1.4110E-05

5 ConclusionIn this paper a new structure of power system stabilizer

to improve small signal stability is introduced. The appli-cation of this PSS is applied to single-machine infinite-buspower system and two-area four-machine ten-bus power sys-tem and, moreover, the performance is compared to the newlyintroduced PSS structure in [3] and without PSS. It is estab-lished that the performance with proposed PSS is highly en-couraging and found better as compared to PSS by Falehi.The results are incorporated in terms of settling time and per-formance indices (ITAE, IAE and ISE) found as least withproposed PSS as compared to PSS structure in [3].

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